improvement of trajectory synthesizer for eﬃcient …...improvement of trajectory synthesizer for...

Improvement of Trajectory Synthesizer for Efficient

Descent Advisor

Min Xue∗

University of California at Santa Cruz, Moffett Field, CA 94035

Heinz Erzberger†

NASA Ames Research Center, Moffett Field, CA 94035

The Efficient Descent Advisor (EDA) is being developed to provide a decision supporttool for controllers to help them issue continuous descent trajectories for arrival traffic.This paper investigates methods for improving the accuracy of the trajectory synthesizer,which is the trajectory engine that supports EDA. The study was motivated by the needto harmonize the trajectories generated by the trajectory synthesizer with those generatedby on board Flight Management Systems (FMS’s), which are used by pilots to executecontinuous descents. An analysis of error sources between predicted and actual continuousdescent trajectories shows that thrust and descent speed profiles are the most importantparameters affecting the accurate prediction of top of descent location and arrival fix cross-ing times. While the thrust correction applies to all descents, the descent speed appliesto uncontrolled (non-metered) descents. In order to establish the best value for thrustcorrection for use in the trajectory synthesizer, a limited set of continuous descent trajec-tories flown by aircraft during regular revenue flights into the Dallas-Fort Worth Airportwere recorded and analyzed. In addition to recorded trajectories, controllers also obtainedFMS-calculated top-of-descent ranges, crossing times and speed profiles from pilots when-ever possible. In post flight analysis the predicted trajectories generated by the TrajectorySynthesizer were compared to the flight test data. It was generally found that the actual(FMS guided) trajectories were flown at shallower descent angles than the predicted tra-jectories and that actual descent speeds often differed significantly from those programmedinto the trajectory synthesizer. A descent thrust correction parameter, normalized to air-craft weight and dependent on aircraft type and airline/operator is introduced to improvetrajectory prediction accuracy. It is shown that this parameter, together with updatedspeeds obtained from pilots prior to descent, increase the accuracy of trajectory predictionsignificantly.

I. Introduction

Continuous Descent Arrival (CDA)1–3 refers to a procedure that allows aircraft to approach an airportfrom cruise altitude at near idle engine power. Compared with typical air traffic arrival procedures, whichcan include multiple level flight segments prior to reaching the crossing altitude at the arrival fix, CDAreduces fuel consumption, emissions, and noise, thus providing both economic and environmental benefits.

Continuous descent trajectories are typically computed by the Flight Management System (FMS) andare executed automatically by an autopilot or flown manually by the pilot following flight director guidance.At low traffic levels controllers can safely accommodate pilot requests for CDA’s. However, as the trafficdensity increases controllers find it increasingly difficult to handle CDA requests without the help of decisionsupport tools. The Efficient Descent Advisor (EDA)3,4 is a tool that has been designed to help controllersmanage CDA’s safely even during busy traffic periods. EDA accomplishes this by computing CDA solutionsthat conform with time-based metering schedules computed by Traffic Management Advisor (TMA)5,6 forbalancing traffic demand and capacity. By following advisories generated by EDA controllers are able to

∗Research Scientist, University Affiliated Research Center. Mail Stop 210-8. AIAA senior member†Senior Advisor, Mail Stop 210-10. email: [email protected], AIAA Fellow

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American Institute of Aeronautics and Astronautics

issue CDA clearances while maintaining high traffic flow and avoiding frequent conflicts. A key element inEDA is a function for accurately modeling FMS-generated descent profiles. This function is performed bythe Trajectory Synthesizer (TS), which is a component shared between TMA and EDA.

This paper focuses on methods to improve the prediction accuracy of TS. A sensitivity analysis was firstconducted to gain an understanding of error sources and their impact on prediction accuracy. The significanceof error sources was prioritized based on error magnitudes observed in actual operations. Then methods ofintroducing updated descent calibrated air speed (CAS) for non-metered flights and a thrust correctionfactor for all flights were proposed for improving descent trajectory predictions. It should be noted that theapplication of adjusting CAS is limited to the uncontrolled arrivals or any prediction computed prior to anyEDA control. Trajectories from flights that executed the continuous descents into the Dallas Fort-Worthairport were collected for examining the improvements. It is shown that the proposed methods have thepotential to reduce the descent trajectory prediction errors to acceptable ranges, thereby improving theprediction accuracy of TS and the performance of EDA.

II. Model of Continuous Descent Trajectory

Typical vertical profiles of CDA’s are shown in Fig. 1. Fig. 1(a) and 1(b) present the spatial and temporalaltitude profiles. And Fig. 1(c) and 1(d) show the corresponding speed profiles. During continuous descent,an aircraft starts from its cruise altitude and descends with a constant Mach (if the desired descent CAS isgreater than the final cruise CAS) until it reaches the pre-defined CAS (point B in Fig. 1(a)). After that,the aircraft continues its descent with the CAS (from point B to C in Fig. 1). Before it reaches the meterfix, the aircraft decelerates to the final CAS. Typically, an aircraft is required to cross the meter fix at analtitude of 10,000 feet with an indicated airspeed of 250 knots, while in Fig. 1 11,000 feet was required.

A trajectory model for calculating descent profiles was developed in the 1980’s.7,8 It typically consistsof three segments, an acceleration/deceleration segment to a specified Mach number, followed by a constantCAS segment, and then a deceleration segment. This model not only provides convenience for pilots andcontrollers but also simplifies the equations of motion to first order differential equations:8

VT =T −D

m− g sin γa − uw cos γa (1)

x = VT cos γa + uw (2)

z = VT sin γa (3)

where T and D are thrust and drag, respectively. VT is the true airspeed, γa is the aerodynamic flightpath angle, and uw is the horizontal component of wind and g is the acceleration of gravity. It is assumedthat there is no vertical wind. x and z are the horizontal and vertical axis in an earth fixed coordinatesystem, respectively. From Eqn. 1, γa for the constant Mach and constant CAS segments can be expressedas follows, respectively:

γa(constantM) =T −D

m[g + VT (M

da

dz+

duw

dz)]−1 (4)

γa(constantCAS) =T −D

m[g + VT (

dVT (VCAS , z)dz

+duw

dz)]−1 (5)

where a is the speed of sound, M is the Mach number, and VCAS is the indicated airspeed. For idle-thrustdescent, γa can be obtained by setting T to idle value. Forward and backward integrations are then usedfor the first and third segments until the conditions of constant indicated airspeed segment are met. Thenthe complete descent trajectory can be obtained by integrating the segment of constant indicated airspeed.More details of descent calculation can be found in previous works.7,8

III. Sensitivity Study of Trajectory Synthesizer

Several trajectory sensitivity studies9–14 have been conducted in past years. Most of those examinesensitivities for error in arrival times and along-track position. The purpose of this section is to identifythe major factors that affect the prediction of descent trajectories in TS. To facilitate comparisons between

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Figure 1. Continuous descent arrival: (a) Spatial profile of altitude (b) Temporal profile of altitude (c)Indicated airspeed (d) Mach Number

trajectories, the descent trajectories are usually treated as two dimensional with horizontal distance to themeter fix and altitude as the two axes.

In this section, the distance from the top of descent (TOD) to the meter fix is called TOD distance, andthe time that the aircraft flies from the reference point to the meter fix is recorded as meter-fix crossingtime. In this section, three typical error sources — descent weight, wind error, and descent speed — areexamined. Without loss of generality B737-800 is used as an example.

A. Descent Weight

Aircraft weight is one of the major concerns in trajectory prediction, especially in the case that datalink is notavailable or airlines choose not to share the weight. Fortunately, the range of descent weights is much narrowerthan the range of take-off weights. In nominal situation, assuming descent fuel consumption is negligible,the minimum descent weight can be estimated using Eqn. 6, where WOWE is the Operating Weight Empty(OWE), and Fres represents minimum reserve fuel required by Federal Aviation Administration (FAA). Thereserve fuel is for cruising to the alternate airport and for 45 minutes of airborne holding. Generally thereserve fuel is set to eight percent of the takeoff weight.15,16 Assuming no extra fuel would be carried by theaircraft, the maximum descent weight is estimated by Eqn. 7, where WMPLD is the maximum payload allowedfor the aircraft, and WMLW is the maximum design landing weight which should not be exceeded. UsingB737-800 as an example, the minimum descent weight of B737-800 is about 106,457 lbs, and the maximumdescent weight is 146,300 lbs, which corresponds to a load factor of 100%. If an aircraft is executing a long

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range flight, the maximum possible descent weight should be further reduced, which means the load factorwill be less than 100%. Currently, in TS, the default descent weight of B737-800 is 126,720 lbs, correspondingto a load factor of 50%.

Wmin = WOWE + Fres (6)

Wmax = min{WOWE + Fres + WMPLD,WMLW } (7)

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typical descent weight (126,720 lbs)minimum descent weight (106,457 lbs)maximum descent weight (146,300 lbs)

Meter Fix

Top ofDescent

Figure 2. Impacts of weight on descent trajectory

10 8 6 4 2 0 2 4 6 8

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ght (

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Fac

tor(%

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(b)

Figure 3. Impacts of weight on (a) TOD distance (b) Meter-fix crossing time.

Figure 2 shows the different descent trajectories associated with different descent weights for CAS 280and standard day without wind. The middle trajectory corresponds the typical descent weight used by TS,the steep one is based on minimum descent weight, and the shallow profile results from maximum descentweight. It is noted that heavily loaded aircraft fly shallower descents with idle thrust than do lightly loadedaircraft. Figure 3(a) shows the difference in TOD range can be as large as 16 nmi from empty payload (loadfactor 0) to full payload (load factor 100%), which means TOD will be 4.1 nmi further from meter fix foreach 10,000 lbs increase in weight. The default weight in the TS is shown as a red dot in the figure, whichdenotes load factor 50%. The difference in meter-fix crossing time over the weight range is small as shown inFig. 3(b). To serve the purpose of this paper, it is assumed that the requirement of meter-fix crossing timeaccuracy is 30 seconds. For the entire range of descent weights, the difference is only about 20 seconds. Foreach 10,000 lbs increase in weight, the increase in meter-fix crossing time is less than 5 seconds. According

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to the analysis conducted by the International Air Transport Association (IATA), the average load factorfor commercial airlines is about 70%. This further narrows the range of weight and reduces the differencein TOD distance to around 3 nmi. Thus, the prediction errors might be tolerable given the fact that theaircraft descent weight can be well estimated. Since even in actual operations, it is unlikely to obtain aircraftweight information, it is useful to compute accurate descent prediction without weights reported from pilotsor airlines.

B. Wind Speed

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No Wind ErrorHead Wind (10 kts)Tail Wind (10 kts)

Figure 4. Impacts of wind on descent trajectory

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Wind error (knot)

Met

er fi

x cr

ossi

ng ti

me

erro

r (se

c)

(b)

Figure 5. Impacts of wind on (a) TOD distance. (b) Meter-fix crossing time.

Wind speeds can be as high as 150 knots at high altitudes and are therefore critical to the trajectorycalculation. Wind speed error exists due to the inaccurate wind forecast. Although the magnitude of windspeed is high, the error is usually less than 10 knots.14,17 Figure 4 shows the trajectories with different windspeeds. The steep trajectory results from a strong head wind, while the shallow trajectory corresponds toa strong tail wind. Strong head wind shortens the TOD distance to the meter fix. Figure 5(a) presentsthe different TOD distances due to wind errors. It can be seen that the shift of TOD distance caused bywind speed error is small. The error of 10 knots only corresponds to 1.3 nmi difference in TOD distance.Figure 5(b) shows the effect of wind error on arrival time, typically 11 seconds for every 10 knots.

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C. Descent Speed

Descent speed is another important factor for descent trajectory calculation. Usually, descent speeds areselected by airlines or pilots for those un-delayed (non-metered) flights. Fig. 6 presents the impacts ofdescent speed on descent trajectory, and Fig. 7(a) and 7(b) show TOD distances and meter-fix crossingtimes corresponding to different descent speeds. It was found that descent CAS has the dominant impact dueto the length of the segment. A 10 knots difference in CAS causes the differences of 3.2 nmi in TOD distanceand 18.1 seconds in meter-fix crossing time. Cruise Mach number has minimal effect of 2.4 seconds and 0.5nmi per 0.01 Mach. It is noticed that descent CAS in actual operations is different from the default CAS inTS. For instance, the trajectory synthesizer of EDA uses a default descent speed of 280 knots for B737-800, a

whereas flights from the actual operational data, which will be discussed in next section, descended at 310knots. This mismatch alone will cause 9 nmi error in TOD distance and 51 seconds error in meter-fix crossingtime, which makes the trajectory prediction unacceptable. Based on this finding, it is strongly recommendedthat pilot-preferred descent speeds be acquired prior to descent in order to make initial trajectory predictionsacceptable.

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default descent speed (280 kts)low descent speed (260 kts)high descent speed (320 kts)

Figure 6. Impacts of descent speed on descent trajectory

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er fi

x cr

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erro

r (se

c)

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Figure 7. Impacts of decent speed on (a) TOD distance (b) Meter-fix crossing time.

aEDA uses the default speed for the initial, un-delayed trajectory calculation only, once EDA advises a speed profile andcontroller accepts, descent CAS is no longer an error source

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IV. Methods for Improving Prediction Accuracy

Based on the impact on the error magnitude and the accessability of the information, the first recommen-dation for improving un-delayed/initial prediction is to acquire the descent CAS prior to descent. Here it isproposed that the intended descent CAS can be acquired via down linked communication from the aircraftto the air traffic control center. If the intended CAS can NOT be down-linked, the default descent speedsin EDA need to be revised based on the actual operations.

Furthermore, a thrust correction factor, which is dependent on aircraft type and airlines, is proposed tomitigate the TOD prediction errors. The thrust correction factor is referred to as τ in Eqns. 8 and 9. It isdefined as a function of nominal aircraft weight W , and a dimensionless tuning parameter cτ is selected tominimize prediction errors.

VT =T + τ −D

m− g sin γa − uw cos γa (8)

τ = cτ ·W (9)

Before applying the thrust correction, the impact of thrust changes on the trajectories needs to beunderstood. Figure 8 shows the impact of a 1.5% thrust correction factor on descent trajectory. A positivethrust correction factor makes the shallow descent trajectory, and the dash-dot line results from a negativethrust correction factor. Figure 9(a) and 9(b) plot the change in TOD location and meter-fix crossing timeover a range of ±1.5% thrust correction, which roughly corresponds to ±15 nmi in TOD and ±20 secondsfor meter-fix crossing time, respectively.

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Figure 8. Impacts of thrust correction on descent trajectory

V. Experiments and Results

In order to examine above methods, a limited set of continuous descent trajectories flown by aircraftduring regular revenue flights into the Dallas-Fort Worth Airport were recorded and analyzed. Based onthe request from NASA, on Feb. 25-26, 2011, controllers acquired FMS-calculated top-of-descent ranges,crossing times and intended descent CAS from pilots who were willing to fly CDA’s during light traffic. Priorto such requests, controllers determined that flights had enough space for executing CDA’s. There were nopilot briefing or training prior to the short experiment. Meanwhile, the flight track information includingaircraft position, altitude, ground speed, and heading was recorded at the NASA’s North Texas ExperimentalFacility, co-located at the Fort Worth Center. The associated wind forecasts were also recorded.

Using one flight as an example, here is how the experiment was conducted: On Feb. 26, 2011, at 22:29GMT (4:29 pm local time), per controller’s request, the pilot reported that they were 10 nmi prior to theFMS-calculated TOD, their intended descent CAS was 261 knots, and their arrival time to meter fix YEAGR

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1.5 1 0.5 0 0.5 1 1.515

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Chan

ge o

f TO

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stan

ce (n

mi)

(a)

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Thrust adjustment (% of weight)

Cha

nge

of m

eter

fix

cros

sing

tim

e (s

ec)

(b)

Figure 9. Impact of thrust correction on (a) TOD distance (b) Meter-fix crossing time.

was estimated to be 22:57 GMT. At 22:46 GMT, the flight was cleared to descend to YEAGER with finalspeed 250 knots and altitude 9,000 ft, respectively.

Overall 14 flights from two major airlines — Airline A and S — were collected and analyzed. Theycomprised four B737-700 from Airline S, and seven B737-800, two B757-200, and one B737-700 from AirlineA.

A. Speed and thrust corrections

In TS, the default descent CAS for a B737-700 is 280 knots, quite different from 261 knots reported bypilots in Airline S cases. Thus, speed correction was conducted in TS. Fig. 10(a) and 10(b) show spatialand temporal profiles calculated by default TS for flight S101. And Fig. 11(a) and 11(b) show spatial andtemporal profiles computed by TS after the speed correction. The blue curves are radar-recorded actualdescent trajectories and associated blue dots are the positions where pilots reported their TOD, time, andspeed. Blue diamonds and triangles are TOD positions and meter-fix crossing times, respectively, calculatedby the onboard FMS and reported by pilots. Yellow diamonds and triangles are actual TOD positions andactual meter-fix crossing time retrieved from radar-recorded track information. Black curves are trajectorypredictions as calculated by TS. Temporal profiles show that before speed correction, there is about a 50second difference between TS prediction (the low end of black curve) and actual crossing time, and actualand FMS crossing times are very close. After speed correction, the difference between TS calculation andactual becomes negligible for this particular set of flights and aircraft type.

As shown in Fig. 10(b) and 11(b), after speed correction there still exists a significant gap between theTS-calculated TOD locations and FMS/actual TOD locations. Thus a unified thrust correction factor mustbe applied on all four B737-700s from Airline S. The thrust correction parameter τ was manually identified as0.7%. Figure. 12(a) and 12(b) present the temporal and spatial profiles after introducing thrust correctionfor flight S101, which is similar to other Airline S flights as well. It can be seen that the TS prediction ofTOD location has been improved significantly after adjustment. Table 1 lists the statistics for all four AirlineS flights using the same thrust correction factor τ . The prediction accuracy desired for EDA operations isfor TOD and meter-fix crossing time errors to stay within 5 nmi and 30 seconds, respectively. It appearsthat these error rates are achievable with the proposed methods.

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Figure 10. Flight S101 actual trajectory and default TS calculation (a) Spatial profiles (b) Temporal profiles

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Figure 11. Flight S101 actual trajectory and TS calculation with updated CAS (a) Spatial profiles (b) Temporalprofiles

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Figure 12. TS profiles with constant thrust correction for flight S101 (a) temporal (b) spatial

Table 1. TS predictions errors for Airline S B737-700

TOD Error (nmi) Time Error (sec)average std. average std.

Default TS 20.3 16.1 38.9 13.0Speed correction 5.6 3.1 5.4 2.7Thrust correction 1.8 1.5 9.4 7.2

B. Early descent phenomenon

Although five out of ten Airline A flights behaved similarly to Airline S flights and TS predictions for theseflights can be significantly improved by same means, there exists an interesting phenomenon - “early descent”for five Airline A flights. Figure 13 and Fig. 14 present such an example, which shows the comparisons before

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and after applying thrust correction for a B737-800 of Airline A. The constant thrust correction factor is1.6%, which was set for all seven Airline A B737-800s. From the figure, it is noticed that although theTS predicted TOD does not match the actual TOD it is a good match for FMS TOD (the blue diamond).Apparently, pilots didn’t follow their FMS calculations and chose to descend almost 30 nmi earlier thanFMS TOD location. At the end of the early descent segment, the descent trajectory merges with the FMScalculated idle thrust trajectory.

Without a debriefing of the pilots after the experiments, the exact reasons for early descents could not beascertained. However, according to FMS experts, pilots frequently choose the “descend now” option on theFMS before the aircraft reaches the FMS calculated TOD location. The primary reason for pilots selecting“descend now” instead of flying the FMS-calculated idle-thrust descent is to make passengers comfortable.

The early descent phenomenon suggest the need to coordinate the descent procedures of airlines withthe ground-based EDA tool. Then, the TS in EDA can be adapted to correct for early descent procedure,thereby improving prediction accuracy.

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Figure 13. Early descent phenomenon: (a) Spatial profiles with NO thrust correction (b) Spatial profiles withthrust correction

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Figure 14. Early descent phenomenon: (a) Temporal profiles with NO thrust correction (b) Temporal profileswith thrust correction

C. Overall results

Table 2. TS predictions errors for all flights

TOD Error (nmi) Time Error (sec)average std. average std.

Default TS 19.0 12.2 22.1 15.5Speed correction 16.1 11.7 10.1 9.3Thrust correction 7.3(1.7) 8.1(1.4) 7.8 7.1

Table 2 shows the overall comparisons between TS prediction and actual track. In the table, the errorsshown in parentheses are the errors between the TS calculation and the onboard FMS (instead of actual).

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(Recall that, in early descent cases, TS prediction can still match on board FMS calculations.) The tableshows that with speed correction, the meter-fix crossing time errors can be halved to 10 seconds with astandard deviation of 9 seconds. With further thrust correction, the TOD location errors can be reducedfrom 19 nmi to 1.7 nmi (if taking away the impact of early descents), which is similar to what was found inAirline S cases.

D. Discussion

Although it was proposed above that a constant thrust correction factor should be pre-defined based onaircraft type and airlines/operator, the thrust correction can be applied in two different ways: 1) If the FMS-calculated TOD location and meter-fix crossing time can be down-linked from an aircraft through Data-link,then in real-time for an individual flight, a thrust correction factor may be calculated based on the down-linked TOD and crossing time. Therefore, an accurate 4D descent trajectory can be calculated by TS, whichis required for reliable conflict detection and resolution when flying continuous descent approaches; 2) If theFMS-calculated TOD location and meter-fix crossing time can NOT be down-linked from an aircraft, thethrust correction factor can be used in the way proposed in previous sections: For a given type of aircraft anda given airlines/operator, determine a constant thrust correction factor from analysis of a set of previouslyrecorded descent trajectories. Both methods can improve TS prediction accuracy for continuous descents.However, down-linked information has the advantage in accuracy. It has to be acknowledged that the smallsample set is not statistically significant. Additional modification needs to be made in TS, so a large amountof on-line experiments can be conducted in TS for validating the proposed methods. Regarding the earlydescents, while the TS is architected to model multiple descent segments based on different parameters (e.g.thrust, fixed flight path angle, or rate of descent), efforts are needed to identify appropriate parameters forearly descents and to develop methods to obtain such intent.

VI. Conclusion

In order to handle continuous descent procedures during busy traffic conditions, controllers using decisionsupport tools, such as EDA, require accurate prediction of descent trajectories. The prediction functionis performed by the trajectory synthesizer (TS). This paper investigated the sensitivity of TS predictionaccuracy to various parameters and proposes methods for improving its accuracy based on actual operationalflight data.

During the sensitivity analysis, the significance of error sources was prioritized based on error magnitudesin actual operations. Assuming that the prediction accuracy requirements of TOD location and meter-fixcrossing time are 5 nmi and 30 seconds, respectively, it was found that for the range of expected wind speedand weight errors prediction errors remained within acceptable limits. Since aircraft descent weight lieswithin a relatively narrow range in practice, it may be acceptable to use an estimated descent weight if theactual weight is not available prior to descent. The FMS-selected descent speed has a significant impacton arrival time prediction for those non-metered flights and should be correctly entered into TS for eachaircraft-FMS combination prior to descent. It is suggested to acquire the FMS-selected descent speed fortrajectory prediction prior to any EDA speed assignment. At least, the nominal descent speeds in TS needto be calibrated based on actual airline policies. As actual descent trajectories were typically shallower thanTS predicted trajectories even after the descent speed correction, a thrust correction factor was introducedto achieve the desired prediction accuracy.

Data were collected from a limited set of flights that executed the continuous descents into DFW airport.Analysis shows that introducing a constant thrust correction factor and intended descent CAS correction hasthe potential to reduce the descent trajectory prediction errors to acceptable/desired ranges. The descentCAS mainly corrected the meter-fix crossing time and the thrust correction factor mainly improved the TODprediction. These corrections could be achieved by the aircraft down-linking critical FMS parameters prior tothe descent. If only descent CAS can be down-linked, a thrust correction factor parameter can be identifiedbased on aircraft type and airlines/operator for future prediction. If TOD locations and meter-fix crossingtime can also be down-linked from an aircraft, a thrust correction factor could be computed in real time foraccurate 4D trajectory calculations. In both cases, the accuracy of descent trajectories required for reliableconflict detection and resolution will be significantly improved. It is suggested that additional modificationcan be made in TS such that a large amount of on-line experiments can be carried for further validation.

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It was also determined, in order to model early descents, appropriate parameters need to be identified formultiple descent segments in the TS model and methods should be developed to obtain the intents of earlydescents.

VII. Acknowledgement

The authors gratefully acknowledge the contribution of Mr. Paul Borchers of NASA and Mr. KeenanRoach of University of California Santa Cruz at NASA/FAA Northern Taxes Research Station. They coor-dinated with air traffic controllers at DFW airport and collected actual operations data for the flights thatexecuted continuous descent arrivals at the airport.

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